A COURSE 



— IN- 



E;iL,jE:3i5vd:EisrTA.]R"Y" 



Mechanic/il Drawing. 



Entered, according to Act of Congress, in the year 1882, by Wm. A. Pike and 
W. F. Deckek, in the Office of the Librarian o| Congress at Washington. 



T 

.P63 



LIBRARY OF CONGRESS. 



7^ TE 

Shelf..--'P.£.3 



UNITED STATES OF AMERICA. 



\ 



lOURSE- 



ELEMENTARY 



MEGHANIGAL DRAWINS 



By 

Wm. a. Pike and W. F. Decker., 



—OP THE— " 



iNIYERSlTY#OF#®IjM]MDSOTA. 

Minneapolis, 1882. 




MINNEAPOLIS: 
L KiiMBALL & Co., Printers, 25 & 27 Second Street South . 

1882. 



COURSE IN 



DRAWING MATERIALS AND INSTRUMENTS. 

Each Student will require the following Instruments on 
beginning the course, viz.: — 

Half-a-dozen Sheets of Dra-wing Paper, a Dra\iring-board, a 
T-square, a Pair of Triangles, a Hard Pencil, a Right Line Pen, 
a Pair of Compasses Tirith Pen, Pencil and Needle Points, a Pair 
of Plain Dividers, an accurate and finely divided Scale, a piece 
of India Ink, a Rubber, an Irregular Curve, and half-a-dozen 
Thumb-tacks. 

These instruments and materials are all that are absolutely 
required up to the time of commencing tinting and shading, 
when a few other articles will be needed, which will be spoken 
of in their proper place. 

Before purchasing, the following directions about the dif- 
ferent instruments and materials should be noticed. 

Paper. — For all the drawings of this course, use Whatman's 
Imperial drawing paper. It comes in sheets of convenient 
size, and is well adapted to the work of the course. Six sheets 
will be enough up to the time of tinting. 

Drawing-Board. — Great care should be taken to secure a 
good drawing-board. The best boards are those made of 
thoroughly seasoned white pine, one inch thick, with cleats 
at the ends flush with the surface of the board. The most 
convenient size is twenty-three by thirty-one inche!r\ This 
gives a small margin outside of a whole sheet of Imperial 
paper, which is twenty-two by thirty-inches. 

One surface of the drawing-board must be plane, and the 



edge from which the T-square is used must be perfectly 
straight. 

T-Squake — All horizontal lines in the drawings are made 
by the use of the T-square. The T-square should be used 
from the left-hand edge of the board, unless the person is 
left-handed, in which case it should be used from the right- 
hand edge. The upper edge of the blade only is to be used in 
drawing lines. The blade should be at least thirty inches 
long, and about two-and- one-half inches wide. The thickness 
should not be over an eighth of an inch. The head should be 
twelve or fourteen inches long, at least, in order that the blade 
may never be thrown out of line. By sliding the head up and 
down on the straight edge of the board, any number of parallel 
horizontal lines may be drawn. It is very important that the 
upper edge of the T-square be perfectly straight. 

Tkiakgles. — For making all vertical* lines, and all lines 
making the angles of thirt}^ forty -five and sixty degrees with 
the horizontal and vertical lines, triangles are used, sliding on 
the upper edge of the T-square. Two triangles are necessary, 
one forty- five degree and one thirtj^ and sixty degree, as they 
are called from their angles. Each of these triangles has one 
right angle, and either can be used for drawing verticals. It 
is often convenient to have one triangle large enough f,or 
drawing quite long verticals, like border lines; but in lettering 
and in other small work a smaller one is much more convenient. 
It is therefore advisable to get a thirty and sixty degree tri- 
angle that has one of its rectangular edges about ten inches 
long, and to get a forty-five degree triangle much smaller. 

Pencils. — All lines are to be made first with a hard pencil, 
and are afterwards to be inked. It is very important that the 
pencil lines be very fine and even, though they need not be 
very dark. 

Ink will not run well over a soft pencil line, and it is impos- 
sible to do good work without making the lines fine. The 
best pencils for this work are Faber's H H H H and H H H 
H H H or some kind equally hard and even. The H H H H 
is recommended for beginners who are not accustomed to using 
a very hard pencil, but the H H H H H H is harder, and bet- 

* By vertical 4ines are meant lines parallel to the edge of the hoard from which the 
T-square is used, by horizoutal those parallel to the upper edge of the T-square. 



ter adapted for nice work. One of each kind will be amply 
sufficient for the work of the whole course. The pencil should 
be sharpened at both ends, at one end with a common sharp 
round point, and at the other with the lead of about the shape 
of the end of a table knife. The round point is to be used in 
lettering and in other small work, and the flat point in making 
long lines. The flat point will keep sharp much longer than 
a round point. Both points should be sharpened often by 
rubbing them on a piece of fine sand paper or on a very fine 
file. The flat point should always be used in 'the compasses 
with the edge perpendicular to the radius of the circle. 

Right-line Pen. — In selecting a right-line pen care should 
be taken to get one with stiff nibs, curved but little above the 
points. If the nibs are too slender they may bend when used 
against the T-square or triangles, and the result will be an 
uneven line. If the nibs are too open there is danger of the 
ink dropping out and making a blot. If too little curved the 
pen will not hold ink enough; the nibs are more apt to be too' 
open, however, than otherwise. The medium sized pens are 
best adapted for this work. The pen must have a good adjust- 
ment screw to regulate the width of the lines. The pens, as" 
they are bought, are generally sharpened ready for use ; but, 
after being used for a time, the ends of the nibs get worn down, 
so that it is impossible to make a smooth, fine line. When 
this occurs they should be sharpened very carefully on a fine 
stone. In order to have a pen run well two things are neces- 
sary, first the points must be of exactly the same shape and 
length, and both nibs must be sharp. In sharpening a pen, 
therefore, the first thing to be done is to even the points. 
This may be done by moving the pen with a rocking motion 
from right to left in a plane perpendicular to the surface of 
the stone while the nibs are screwed together. After the nibs 
are evened in this way, the points sliouM be opened, and each 
nib sharpened by holding the pen at an angle of about thirty 
degrees with the surface of the stone, while it is moved in 
about the same manner as in sharpening a gouge. The point 
should be examined often with a lens. 

Compasses. — The compasses must have needle points, with 
shoulders to prevent them from going into the paper below a 
certain depth. The needle point, when properly used, leaves 
a very slight hole in the centre of each circle ; while the tri- 



6 

angular point, with which the poorer instruments are provided, 
leaves a very large, unsightlj- hole, unless used with more than 
ordinary care. The pencil point should be one made to con- 
tain a small piece of lead only. All that has been said in 
regard to the right-line pen applies equally well to the pen 
point of the compasses. In using the pen point be sure that 
both nibs press equally on the paper, otherwise it will be 
impossible to make an even line. Both nibs may be made to 
bear equally by adjusting the joints in the legs of the com- 
passes. 

Dividers. — The dividers should be separate from the com- 
passes, as it is very inconvenient to be obliged to change the 
points whenever the dividers are needed. The dividers have 
triangular points, which should be very fine, and of the same 
length. The legs of the dividers should move smoothly in 
the joint, and not hard enough to cause them to spring while 
being moved, The dividers are used principally for spacing 
off equal distances on lines, but are often used for taking 
measurements from the scale, especially when the same 
measurement is to be used on several different parts of a 
drawing. 

Scale. — A very good scale, for this course, is one with 
inches divided into fourths, eights, sixteens, etc., on one edge; 
and into twelfths, twenty-fourths, etc , on the other. The 
first edge is very convenient for taking measurements, and for 
making drawings to a scale of one-half, one-fourth, etc.; but 
the second is better for drawing to a scale of a certain number 
of inches to the foot. Triangular scales are still better, but 
more expensive. 

IiTK. — India ink, which comes in sticks, is the best ink for 
general uses ; but the Higgins' ink, in bottles, is much more 
convenient for making line drawings. None of the ink that 
comes in bottles, however, is good for shading. If the India 
ink is used, an ink slab or saucer will be needed in addition to 
the instruments mentioned in the list. In grinding India ink, 
a small quantity of water is sufiicient,. and the ink should be 
ground until a very fine line can be made very black with one 
stroke of the pen. Ink will look black in the slab long before 
it is fit to use on a drawling. Ink should not be ground, 
however, so thick that it will not run well in the pen. The 



ink must be kept covered up or it will soon evaporate so much 
as to be too thick to run well. 

RuBBEB. — Get a soft piece of rubber so as not to injure the 
surface of the paper in rubbing ; what is known as velvet 
rubber is well adapted to the draughtsman's use. 

Ikregular Curve. — In selecting an irregular curve, one 
should be obtained which has very different curvature in dif- 
ferent parts, in order to fit curves which cannot be drawn with 
compasses. 

Thumb Tacks. — Thumb tacks should have good large heads, 
so firmly fastened on that thej^ cannot get loose. 

One can do much better in buyino- instruments, to buy them 
in separate pieces, each carefully selected, than to buy them 
in sets. It is very difiicult to buy a set of instruments that 
will contain just what is required for this work, without buying 
many unnecessary pieces. 

GENERAL DIRECTIONS FOR COMMENCING THE WORK. 

Each plate of geometrical problems is to be made on a half 
sheet of the Imperial paper. The sheet should be folded over 
and cut with a sharp knife, but before cutting find out which is 
the right side of the paper. The right side of Whatman's 
paper may always be found by holding the sheet up to the 
light. When the name of the manufacturer can be read from 
left to right, the right side is the one toward the holder. 
The half that has not the name on it should be used first, 
while its right side is known ; the right side of the other piece 
can be found in the way described, when it is to be used. 

Place the paper on the drawing board so that two of its 
edges will be parallel to the upper edge of the T-square 
when in position on the edge of the board ; and fasten it down 
with three thumb tacks in each of the long sides, placing each 
thumb tack within a quarter of an inch of the edge, in order 
that the holes may be cut off when the plate is trimmed. For 
convenience in working on the upper part of the plate, it is 
best to have the paper as near the bottom of the board as pos- 
sible. 

Begin each plate by drawing a horizontal line, with the use 
of the T-square, as near the thumb tacks at the top as possible. 
Fourteen inches below the first line, if the longest dimension 



8 

is to be horizontal, draw another parallel to it, at the bottom 
of the paper, and b}' means of the larger triangle, draw ver- 
tical lines at the right and left of the paper, twenty- one 
inches apart. These lines are the limits of the plate, and are 
the ones that the plate is to be trimmed by when finished. 

All of the plates that are to be drawn on a half-sheet must 
be of this size, twenty- one by fourteen inches, unless the 
paper is to be shrunk down, in which case the plates must be 
made somewhat smaller, as will be afterwards noticed. 

All the plates are to have a border line one inch from the 
^nished edge, except on the top, where the border is to be one 
and a quarter inches from the edge. This border should next 
be drawn by spacing off the proper distances from the lines 
just drawn, and drawing the border with T-square and tri- 
angles. 

There are to be eight geometrical problems to each plate, 
and, for convenience in locating them, the space inside of the 
iDorder lines, in the first five plates, should be divided into 
-eight equal rectangles, four above and fonr below a horizontal 
line through the center. These last lines are not to be inked, 
but must be erased when the plate is completed. 

The first five plates in the course are of geometrical prob- 
lems. The problems that have been selected have many 
applications in subsequent work ; and, moreover, the exact 
-construction of them gives the best of practice for beginners 
in handling the different instruments. The construction of 
each problem is described in the text, with references to the 
plates ; and each must be constructed according to the direc- 
tions. The reasons for the different constructions, though 
necessarily omitted in the text, will be evident to every one 
who has a knowledge of phine geometry. 

The geometrical problems are not to be drawn to scale, but 
the}^ should be so proportioned that they will occupy about 
the same amount of space in the centre of each rectangle. 

All of the lines must be made very fine and even ; and great 
care must be taken to get good intersections and tangencies. 

DIRECTIONS FOR LETTERING. 

After the problems are pencilled they must be lettered to 
'Correspond to the plates in this pamphlet. Make all the let- 
ters on the plates of geometrical problems and elementary 



9 

projections, like those given in Plate A. These skeleton let- 
ters are the simplest of all mechanical letters to construct, 
and, when well made, they are more appropriate for such work 
than if more elaborate. Make the small letters, in every case, 
two-thirds as high as capitals. 

Before making a letter draw a small rectangle that will just 
contain the letter, and then construct the letter within the 
rectangle, as shown in plate A, using instruments wherever 
possible. The height of all the capital letters in the problems 
and in the general title at the top, is to be one quarter of an 
inch. The widths vary, and may best be found in each case, 
until practice renders it unnecessary, by consulting plate A. 
Great care must be taken in lettering to make all the lines of 
the letters of the same size, and in joining the curves and 
straight lines. 

TITLES. 

The title of each plate of geometrical problems must corres- 
pond to that given in plate I, except as to number. The titles 
of the projection plates will correspond to that of plate YI, 
except as to number, and the titles of all other drawings will 
be as indicated in the text. In constructing a title always 
work both ways from the central letter of the title, in order 
that the title may be symetrical, and over the center of the 
plate. In order to find the middle letter of the title, count the 
number of letters, considering the space between words as 
equal to that of a letter, and divide the number of spaces thus 
found by two ; this will give the number of the middle letter 
from either end of the title. Construct this letter over the 
center of the plate, and then work both ways from this in the 
way just indicated. Make the letters in a word about an 
eighth of an inch apart, though the space will vary with the 
shape of the letter ; and the space between words equal to 
that of an average letter with its spaces. 

It is best, in all cases, to have the title before you in rough 
letters, to avoid making mistakes in working backwards from 
the middle letter. The titles at the top are to be made in cap- 
itals. The letters in the general title are to be a quarter of an 
inch high and a quarter of an inch above the border, and 
those in number of the plate of letters three-sixteenths of an 
inch hi2:h and the same distance above the s^eneral title. 



10 

The name of the draughtsman should be, in the first seven 
plates, at the lower left-hand corner, three-sixteenths of an inch 
below the border, and the date of completion in a correspond- 
ing position at the right. Make the date first, and commence 
the name as far from the edge, at the left, as the last figure of 
the date comes from the right-hand edge. Make the capitals 
in name and date three-sixteenths of an inch high. 

Number the problems as thej^ are in the plates, commencing 
the first letter of the abbreviations for problems in capitals, 
one-half an inch below, and half an inch to the right of the 
lines forming the upper right hand corner of the rectangle. 
The other letters of the abbreviations are to be small, and 
the numbers of the problems are to be marked with figures 
of the same height as the capitals. 

Great pains must be taken in lettering the plates, as the 
general appearance of a drawing h very much affected by the 
arrangement and construction of the letters and titles. The 
directions here given apply to the plates of geometrical prob- 
lems. Some modifications will be made in lettering the 
problems in projection ; but the remarks on the construction 
of the separate letters, and' on the arrangment of the letters 
in a title, are general. After having had the practice in 
spacing and proportioning the skeleton letters, in the first 
seven plates, the student will be allowed to use other styles of 
letters on the remaining work. Care must be taken, however, 
to have the titles symetrical, and no letters on the plates of 
this course should be made over half an inch high. 

INKING. 

When the lettering is all done, a plate is ready to be inked. 
Before using the pen on the plate, be sure that it is in a con- ^ 
dition to make a fine even line, by testing it on a piece of 
drawing paper, or on the part of your paper that is to be 
trimmed ofF. Be sure to have ink enough ground to ink the 
whole plate, as it is not best to change the ink while working 
on a plate, for the reason that it is nearly impossible to get 
the second lot of the same shade and thickness as the first. 
The arcs of circles should be inked first, for it is easier to get 
good intersections and taugencies by so doing, than it is if the 
straight lines are drawn first. Jfake cell the given lines and cell 
the required lines in full; hut all the construction lines in fine 



11 

clots. Make all the lines in the geometrical problems as fine 
and even as possible. The border lines should be made a little 
heavier than the others. All the fine lines should be made, 
if possible, with one stroke of the pen. In order that an 
even line may be made, the pen must be held so that both nibs 
will bear on the paper equally ; and in order to do this, the 
T-square or triangle must be held a little way from the line, 
but parallel to it. The pen should be inclined slightly in the 
direction it is moved. 

In using the compass pen, the joints of the compass legs 
should be so adjusted that the point will bear equally on both 
nibs. 

The ink should be placed in the pens hy means of a quill or 
a thin sliver of wood. The pen should never be dipped into 
the ink. 

THE PLATES. 

The plates in this pamphlet are given to show the arrange- 
ment and construction of the problems, but should not be 
followed as examples too closely, as mechanical difiiculties 
make it necessary to use coarser lines in proportion to the 
size of the plates than should appear on the drawings. 

GEOMETRICAL PROBLEMS. 

Problem 1. To bisect a given line, A B, or to erect a per- 
pendicular at the middle point of A B. 

From A and B as centres, with a radius greater than one- 
half of A B, described two arcs intersecting at C, and two 
arcs intersecting at D. Join C and D by a straight line, it 
will bisect A B, and will be perpendicular to it. 

Vroh. 2. To divide a given line, A B into anj^ number of 
equal parts, five for instance. 

Draw a line, A C, making any angle with A B, and on A C 
set ofi* any five equal distances, A 1, 1 2, 2 3, 3 1: and 4 C ; join 
C and B, and through 1, 2, 3 and 4, draw lines parallel to C B, 
these lines will divide A B into equal parts. 

Proh. 3. To draw a perpendicular to a line B C, from a 
point A, without thfe line. 

From A as a centre, and with any radius, describe an arc, 
cutting B C in D and E. From D and E as centres, describe 
two arcs intersecting in F. Join F with A. 



Proh '4. To draw a perpendicular to a line B C from a I 

point A, nearly over one end, C, so that problem 4 cannot 
be used. 

From any point B, on the given line as a centre, describe 
an arc passing through A. From some other point D, of B C 
describe another arc passing through A. Join A with the 
other point of intersection of the arcs. 

Prob 5 To erect a perpendicular to a line B C, at a given 
point A, of the line. 

Set off from A, the equal distances A E and A F, on either 
side. From E and F as centres, with any radius greater than 
A Ji and A F, describe two arcs, intersecting at D. Join 
D with A. 

Proh. 6. To erect a perpendicular to a line A E, at a point 
A, at or near one end of A E, where problem 5 cannot be used. 

i^rom some point C, outside of A E, as a centre, and C A as 
a radius, describe an arc cutting A E in A and B. Draw a ' 
.ine through B and C, and produce it till it meets the arc again 
in L». Join D with A. 

Prob 7. To draw a line parallel to a given line A B, at a 
given distance from A B. 

From two points C and D, of A B, which should not be too 
near together, describe two arcs, with the given distance as a 
radius Draw a line E P tangent to these arcs. 

Prob. 8. Through a given point A, draw a line parallel to 
a given line B C. i v 

t ^'^n \lrt ''^'^^''^' ''''"''■'''^ *" ''•■'= "'^"Ch shall be tangent 
to B C. With some point, P, of B as a centre, and with 
the same radius as before, describe an arc. Draw throu<.h A 
a ime tangent to the last arc. 

Prob. 9. To lay off an angle, at a given point a, on a given 
line a c, equal to a given angle BAG. 

With A as a centre and any radius, describe an arc included 
between B A and A C. With a as a centre and the same 
radius, describe an indefinite arc. Lay off the chord b c equal 
B C from c on the arc b c. Join b with a. 

• ^?u' ■'f; 7" ^'^^"^ '' ^"""^ ^^'^'^ BAG, whose vertex A 
is within the limits of the drawing. « 

From A as a center describe an are. cutting A B and A C in 
b and a respectively. From b and a as centres describe two 
arcs intersecting in c. Join c with A. 



13 

Proh. 11. To bisect an angle B A — C D, wliose vertex is 
not within the limits of the drawing. 

Draw by Problem 7, two parallels, a b and a c to A B and C 
D respectively, and at the same distance from A B and C D ; 
this distance must be such that a b and a c shall intersect. 
The problem is then reduced to one of bisecting b a c, which 
is done by problem 10. 

Proh. 12. To pass the circumference of a circle through 
three points, A, B, C. 

Draw the lines A B and B C. Bisect A B and B C by per- 
pendiculars, by problem 1. With D, the intersection of these 
perpendiculars, as a centre, and D A as a radius, describe a 
circumference, it will pass through A, B and C. 

Proh. 13. To draw two tangents to a circle, whose centre 
is 0, from a point A, without the circle. 

Join A with ; on A as a diameter, describe a circle. 
Join the points B and C, in which the latter circle intersects 
the given one, with A. A B and A C will be the required 
tangents. 

Proh. 14. To draw circles, with given radii, c and d, tan- 
gent internally and externally respectively to a circle whose 
centre is 0, at a given point A. 

First, internally. Join A with 0. Lay off on A from 
A, A a equal to c. From a as a centre, and A a as a radius, 
describe a circle. 

Second, externally. Prolong A, and lay off from A, A b 
equal to d. With b as a centre and A b as a radius, describe 
a circle. 

Prob. 15. To draw a circle with a given radius m, tangent 
to two given lines A B alid A C. 

Draw a b and a c parallel to A B and A C respectively, and 
at a distance from them equal to m. With the point of inter- 
section a, of a b and a c as a centre, and m as a radius, 
describe a circle. 

Proh. 16. To draw a circle with a given radius m, tangent 
to a given circle 0, and to a given line, A B. 

With as a centre and a equal to m plus the radius of the 
given circle, as a radius, describe an arc, a b. Draw the line 
C c parallel to A B,and at a distance m from it, by problem 7. 
With C, the intersection of the arc and parallel, as a centre, 
and m as a radius, describe a circle. 



14 

Proh. 17. To draw a circle tangent to a given circle 0, at a 
given point A, and to a given line B C. 

Join A ; at A draw the tangent A B, perpendicular to 
A, and produce it till it meets B C at B. Bisect the angle A 
B C, by the line B a, by problem 10. Produce B a till it 
meets A produced in D. With D as a centre, and D A, a 
radius, describe a circle. 

Proh. 18. To draw a circle, tangent to a given line A B, 
at a given point C, and to a given circle 0. 

At C, draw D C, perpendicular to A B, by problem 5, and 
produce D C below A B till C a is equal to the radius of the 
given circle. Join a with 0, and by problem 1, erect a perpen- 
dicular D b at the middle point of a. W ith the intersection 
D, of D a and D b, as a centre, describe a circle. 

Proh. 19. To draw a circle tangent to a given circle C, at a 
given point A, and to a given circle 0. 

Join A C, and produce it till A D is equal to the radius of 
the other circle. Join D with 0, and bisect D hj a perpen- 
dicular E a, by problem I. With E, the intersection of E a 
and A D produced as a centre, and E A as a radius, describe a 
circle. 

Proh. 20. Given two parallels, B A and C D, to draw a re- 
versed curve which shall be tangent to them at A and C. 

Join A C. Bisect A C in 2, which will be the reversing 
point. Bisect A 2, and 2 C by perpendiculars, 1 F and 3 E. 
Draw A F and C E perpendicular to B A and C D, and with 
the intersection E of C E and 3 E, and the intersection F of 
1 F^ and F A, as centres, and radii, equal to E C or A F, 
describe two arcs. 

Proh. 21. Given two parallels, B A and C D, to draw a 
reversed curve, whose tangents at A and C shall be perpendic- 
ular to B A and C D. 

Join A C Bisect A C in 2, which will be the reversing 
point. Bisect A 2, and 2 C by perpendiculars 1 E and 3 F. 
With Uie intersection E, of 1 E and B A, and the intersec- 
tion F, of 3 F and D C^ as centres, and radii equal to E A or 
F C, describe two arcs. 

Proh. 22. To divide a given line A B in extreme and mean 
ratio. 

Bisect A B by problem 1. At one extremity B, erect a per- 
pendicular B C, and lay ofiP on it B C, equal to one-half of A 



i 



15 

B. With C as a centre and B C as a i%dius, describe an arc, 
cutting A C at D. With A as a centre, and A D as a radius, 
describe an arc, cutting A B in E. Then E will divide A B in 
extreme and mean ratio. 

Prob. 23. Given a circle 0, to inscribe and circumscribe 
squares. 

Through draw two diameters, B C and A D, perpendicular 
to each other. Join their extremities, A, B, C and D, for the 
inscribed square. To circumscribe a square, draw lines through 
A and D parallel to B C, and through B and C, parallel to 
AD. 

Prob. 24. Given a circle 0, to inscribe and circumscribe 
regular hexagons, and to inscribe a regular triangle. 

Lay off the radius A, six times as a chord on the circum- 
ference, for the inscribed hexagon. For the circumscribed 
hexagon, draw parallels to the sides of the inscribed figure, 
which shall be tangent to the circle. 

Join the alternate points of division of the circle for the 
inscribed regular triangle. 

Prob. 25. Given a circle 0, to inscribe and circumscribe 
regular octagons. 

By problem 23, obtain the sides A B, and so on of an in- 
scribed square. Bisect these chords hj perpendiculars, and 
thus the arcs subtended by them. Joins the points C, &c., 
with the vertices of the square for the inscribed octagon. 
For the circumscribed octagon, proceed as in circumscribing 
the regular hexagon in problem 24. 

Prob. 26. Given a circle 0, to inscribe and circamscri'ze 
regular decagons. 

By problem 22, divide the radius B, in extreme and mean 
ratio at b. Apply the larger portion b, ten times to the 
circumference as a chord. Circumscribe a regular decagon 
from the inscribed, as in the last two problems 

Prob. 27, Given a circle 0, to inscribe and circumscribe 
regular pentagons. 

Obtain the vertices of an iuscribed regular decagon hy 
problem 26, and join alternate vertices. Proceed for the cir- 
cumscribed pentagon, as usual. 

Prob. 28. To construct a regular polygon with a given 
number of sides, five for instance, the sides to be of a given 
length A B. 



16 

On A B as a radius describe a semi-circle. Divide the 
semi-circumference into five equal parts, A 4, 4 3, and so on. 
Omitting one point of division 1, draw radii through the re- 
maining points and produce them. AVith 2 as a centre, and 
A B as a radius, describe an arc cutting B 3 produced in C ; 
B 2 and 2 C will be two sides of the polygon. With C as a 
centre, and A B as a radius, describe an arc cutting B 4 pro- 
duced in D; CD will be another side. Continue this construc- 
tion; the last point should come at A. 

Pfoh. 29. On A B and C D as major and minor axes, to 
construct an ellipse. 

We proceed on the principle that tho sum of the distances 
of any point of an ellipse from the foci is equal to the major 
axis. We must first fix the position of the foci. From C as 
a centre, and B as a radius, describe two arcs, cutting A B 
in a and b, these are the foci. To apply the principle just 
mentioned, take the*distances from any point, as c of A B to 
A and B as radii, and a and b as centres. By describing arcs 
above and below A B, and using both radii from each centre, 
four points of the ellipse will be obtained. Other points are 
obtained by taking other points on A B, and proceeding in the 
same way. Connect the points found in this way by using the 
irregular curve. In using the irregular curve always be sure 
to have it pass through at least three points. 

Prob. 80. On A B and C D as major and minor axes, to 
construct an ellipse ; another method. 

On the straight edge of a slip of card board or paper, set off 
three points o, c, a, the distance o a being equal to the given 
semi-major axis, and o c to the semi-minor. Place the slip in 
various positions such that a shall always rest on the minor, 
and c on the major axis. The various positions marked by 
the point o will be points of the ellipse. 

Proh. 31. To construct a parabola, given the focal distance 
E. 

We proceed on the principle that the distance of any point 
from a line A B, called the directrix, is equal to its distance 
from a certain point called the focus. Draw the indefinite 
line A B, for the directrix, and C D perpendicular to it. From 
C, lay off C E and E each equal to the focal distance. The 
point is the focus. Draw a number of perpendiculars to C 
D at various points. To find the points in which the parabola 



intersects any one of them as a a', describe an arc with as a 
centre, and a C, the distance from that perpendicular to C as a 
radius. E is the point of C D through which the curve will 
pass, 

Proh. 32. To construct a parabola between the points C 
and A on two given lines C B and A B, and tangent to the 
former. 

Divide C B into any number, as five, equal parts, C a, 
a b, b c, &c., and B A into the same number of equal parts, 
B a', a' b', &c. Connect C with a', b', c', d' and A. Draw from 
a, b, c and d, lines parallel to A B. The points of the curve 
will be those in which the parallel from d meets C d', the 
parallel from c meets C c', that from b, C b', and so on. 

Proh. 33. To construct a hyperbola, having given the dis- 
tances A and a 0, on the horizontal axis, from the centre 
to either vertex, and from the centre to either focus. 

In the hyperbola, the difference of the distances of any 
point from the foci is equal to the distance between the ver- 
tices, as in the ellipse the sum. Lay ofp from the equal 
distances AO and B to the vertices, and the equal distances 
a and b to the foci. To obtain any point of the curve, 
take any point on the axis as c ; with c A and c B as radii, and 
a and b as centres, describe four pairs of intersecting arcs, as 
in the ellipse ; the points of intersection will be points of the 
hyperbola. By taking other points on the axis, other points 
of the curve will be obtained in the same manner. 

Proh. 34. To construct a curve similar to a given curve B 
A C, and reduced in a given ratio, one-half for instance. 

Draw some indefinite line, a centre-line, if possible, in the 
given curve, as A D. On A D, lay oflP a number of distances ; 
at the points of division, erect perpendiculars to meet the 
curve above and below A D. Draw an indefinite line a d, and 
on it lay off" distances bearing respectively to those laid off on 
A D, the given ratio. Through these points of division draw 
perpendiculars, and lay off on them above and below a d, dis- 
tances bearing the given ratio to those on the perpendiculars 
to AD. 

Proh. 35. To describe a given number of circles, six, for 
instance, within a given circle 0, tangent to each other and 
to the given circle. 

Divide the circumference of the siven circle into twice as 



18 

many parts as the number of the circles to be described, 1 2, 
2 3, &c. To obtain the first circle draw a tangent to the given 
circle at 1, 1 a ; produce 2 to 1 a, at a. Lay off on a, from 
a inwards the distance 1 a to b, since tangents to a circle are 
of equal length. At b draw a perpendicular to a, meeting 
1 in c. With c as a centre and c 1 as a radius describe a 
circle. From as a centre and c as a radius, describe a 
circle intersecting the alternate radii 3, 5, &c., in points 
which will be centres of the required circles. 

Proh. 36. To construct a mean proportional to two given 
lines A D and D B. 

With the sum of these lines as a diameter, describe a semi- 
circle A C B. At the point D, between the two lines, erect a 
perpendicular, meeting the circumference at C. DC will be 
the mean proportional required. 

Proh. 37. To divide a line a b into the same proportional 
parts as a given line A B is divided by the point C. Draw a b 
parallel to A B, and draw lines through A and a, B and b, till 
they meet in d. Draw C d, the point c will divide a b into 
the same proportional parts as C divides A B. 

Prob. 38. To draw a tangent to the ellipse 0, at a given 
point A. Find the foci F and F' as in problem 29 Draw A 
F and A F' and bisect the angles between these lines by prob- 
lem 10. Draw A C at right angles to A B. It will be the 
required tangent 

Proh, 39. Given a square A B C D, to cut off the corners 
so as to form a regular octagon: 

Draw the diagonal A C. From A, lay off' A a, equal to A 
D. From a draw a c and a b, parallel to the sides of the 
square. Join b c, which will be one of the sides of the 
octagon. To obtain other vertices, lay off from the other 
vertices of the square distances equal to C c. 

Proh. 40. To draw a circle, tangent to three straight lines, 
A B, BC, and CD. 

Bisect the angles B and C, by problem 10. Produce the 
bisecting lines till they meet at E. With E as a centre and 
the distance to either line as a radius, describe an arc. 

A special case of this problem, when the three lines form a 
closed figure, gives a circle inscribed in a triangle 



1 



19 
PROJECTIONS. 



If we wish to represent a solid body by drawing, and, at the 
same time, to show the true dimensions of that body, we must 
have two or more views, or projections, of it on as many 
different planes. Take for example a cube. In order to show it 
in a drawing, we must have views of more than one face, in order 
to show that the body has three dimensions. We will consider 
this cube to be behind one plate of glass and below another, 
and in such a position that two of its faces are parallel to these 
plates, which are respectively vertical and horizontal. Now 
suppose that perpendiculars are dropped from every corner of the. 
cube to each of these plates. The points where these perpendic- 
ulars pierce the surfaces of the plates, are called, respectively, 
the vertical and horizontal projections of the corners of the 
cube. If these points be joined by lines, correspondiag to the 
edges of the cube, we shall have in this case, exact figures of 
the two faces of the cube that are parallel to these plates. 
These two figures are called, respectively, the vertical and hor- 
izontal projections of the cube, according as they are on the 
vertical or horizontal plates. In this way we may get two 
views of any solid object, supposing it to be in such a position 
as that of the cube, in the case just noticed, with reference to 
two plates of glass, which we will now call the vertical and 
horizontal planes of projection. 

If the object has a third side very different from the two 
shown in this way, we may consider it to be projected on a 
third plane perpendicular to the two others, and on the side of 
the object to be represented. 

A fourth side may in the same way be represented on a fourth 
plane; but three projections are generally all that are needed to 
show even very complicated objects; and in most cases two 
projections, a vertical and a horizontal, are all that are neces- 
sary, lines on the opposite faces being shown b}^ dotted lines 
on the projections of the faces toward the planes.* 

* We have considered the planes of projection to be in front and above the object to 
be represented, but drawings are often made with the planes behind and below. It is, 
however, believed that the method given is better for practical use. Details are ofteai 
shown as projected on oblique planes, as planes parallel or at right-angles to the axis of 
an object. 



20 

As it is not convenient to have two or more separate dry- 
ings of an object on different planes, as w^ould be necessary if 
we were to represent the projections of the body in their true 
positions; we may consider that the body has been projected 
in the manner indicated, and that the planes of projection 
have been revolved about their intersections so as to bring 
them all into the horizontal plane, with the end views, if any, 
on the right and left of the vertical projection, and the horizon- 
tal projection above the vertical. In this way we may bring all 
the different views or projections into the plane of the top 
surface of the drawing paper ; and by representing the inter- 
sections of the planes of projection by lines, we may show all 
the projections in their true relative position in one drawing. 

The line that represents the intersection of the vertical and 
horizontal planes of projection, is called the ground line. The 
ground line, as well as the other lines of intersection of the 
planes of projection, is often omitted in actual drawings. 

It will appear on consideration of the method of projection 
that the distances of the projections of any point from the 
ground line show the true position of the point in space, with 
reference to the planes of projection. Suppose, for example, 
the horizontal projection of a point to be one inch above the 
ground line, and the vertical projection to be two inches 
below the same line, this shows that the true position of the 
point in space is one inch back of the vertical plane and two 
inches below the horizontal plane. Moreover, it may easily 
be demonstrated that the two projections of a point always lie 
in a common perpendicular to the ground line. 

As lines are determined by locating points in them, the 
principles just given apply in getting the projections of any 
figure that can be represented by lines. [n the problems in 
projection, following, the ground lines must be drawn and the 
points located in the manner just indicated. 

An object may be in any position whatever with reference 
to the planes of projection; but for convenience the body is 
usually considered to be in such a position that the vertical 
projection will show the most important view of the object, 
such, for example, as the front of a building. 

In representing an object of this kind in projection, the front 
of the object is usually considered parallel to the vertical plane 
of projection. 



21 

The vertical projection of an object is called its elevation^ 
and the horizontal projection, its plan. The other projections 
are called e^^(i viewSi or sections, according as they represent an 
end or some part cut by a plane passing through the object. 

By the method of projection just explained each projection 
represents the view of the object a person would have were 
the eye placed on the side of the object represented by the pro- 
jection and at an infinite distance from it. When an object 
is viewed from a finite distance it is seen in perspective and 
not as it really is. Projections show an object as it really is, 
and not as it appears in perspective. Projections are there- 
fore used to represent bodies in their true form and are em- 
ployed as working drawings, in which a body to be con- 
structed is represented as it would appear in projection when 
finished. j 

Plate B shows the projections of a cubical object having twa 
slots in its front face, a rectangular depression in the top, a 
cylindrical depression in the right hand face, a cylindrical pro- 
jection on the left and a slot running nearly through from the 
bottom on the back face. A is the elevation, B the plaUy C the 
left hand end view, D the right hand end view, and E the bottom 
view. 

This plate is not to be drawn, but is given to illustrate this 
method of projection. By a a careful study of this plate it may 
readily be seen how the lines of an object will appear in the 
several projections. 

LINE SHADING. 

In order to give the projections of a body the appearance of 
relief, the light is supposed to come from some particular 
direction, and all lines that separate light faces from dark 
ones are made heavy. 

The direction of the light is generally taken for convenience 
at an angle of forty -five degrees from over the left shoulder as 
the person would standnn viewing the projections, or in mak- 
ing the drawing; and in all cases the projections of this pam- 
phlet are to be shaded with the light so taken. 

It will be readily seen on considering the direction of the 
light, that the elevation of a solid rectangular object with 
plane faces in the common position, will have heavy lines at 



22 

the lower and right hand sides, and that the plan will have 
heavy lines on the upper and right hand sides. 

When a body is in an oblique position with reference to the 
planes of projection, the heavy lines of the projections may be 
determined by using the forty-five degree triangle on the T- 
square. If we apply this triangle to the T-square, so that one 
of its edges inclines to the T- square at an angle of forty-five 
degrees upward, and to the right this edge will represent the 
horizontal projection of a ray of light; and by noticing what 
lines in the plan of the object this line crosses, it may readily 
be seen what faces in the elevation will receive the light and 
what faces wi' *»be in the shade. By applying the triangle so 
that an edge will make an angle of forty-five degrees down- 
ward and to the right, this edge will represent the vertical pro- 
jection of a ray of light, and by applying it to the elevation, 
the faces in the plan that will be in light and shade may be 
determined. 

Where the limiting line of a projection is an element of a 
curved surface as in the elevation of a vertical cylinder, that 
line should not be shaded. The plan of the vertical cylinder, 
which is a circle, should be shaded, for the circumference is an 
edge separating light from dark portions of the object. In 
this case the darkest shade of the line should be where the 
diameter, that makes an angle of forty-five degrees to the 
right with the T-square cuts the circumference above, and the 
lightest part should be where this diameter cuts the circumfer- 
ence below. The dark part should taper gradually into the 
light part. 

Lines that separate parts of a body that are flush with 
each other, as in joints, should never be shaded, and 
when a line that would otherwise be shaded, rests on a hori- 
zontal plane as in the first positions of the following problem, 
it should not be shaded. 

The shaded lines of a projection need not be very heavy if 
the light lines are made, as they should be, as fine as possible. 

Plate B is shaded correctly for the light at forty-five degrees 
from over the left shoulder, and a study of it will illustrate all 
that has been stated as to the proper lines to be shaded and 
the relative widths of the light and heavy lines. 



23 

THE PLATES OF PROJECTIONS. 

The plates of projections are to be of the same size as those 
of geometrical problems, but for convenience in showing three 
positions of each object adjacent to each other, the top of these 
plates will be taken at one of the short edges, and each plate 
will contain nine problems in the position shown in the cuts. 
The border should be drawn first, leaving a margin of one 
and a quarter inches at the top as in the previous plates. 
Divide next the space within the border line into nine equal 
rectangles, by drawing two vertical and two horizontal lines. 
Draw a ground line in each rectangle two inches and a half 
below the top, making the ends of the ground lines within a 
quarter of an inch of the vertical lines dividing the space. At 
a distance of two inches and three-quarters below the ground 
line draw a broken horizontal line, as shown in the plates. 

The objects projected in the following problems are all sup- 
posed to rest on a horizontal plane below the horizontal plane 
of projection and behind the vertical plane of projection. The 
ground line, as has already been noticed, represents the inter- 
section of the two planes of projection before the vertical plane 
is revolved into the plane of the horizontal. The broken line 
below represents the intersection of the vertical plane with the 
plane on which the body rests. 

In order to get a good conception of the position of the 
object, suppose that the body rests on the drawing table, and 
that a plate of glass be held above and parallel to the table, 
and another plate be held in front and vertical. The posi- 
tion of the object in relation to the planes represented by the 
plates of glass will be the same as that of the cube, which we 
considered in explaining projections in general. 

The ground line and the broken line below will represent in 
this case, respectively, the intersections of the plates of glass 
with each other and with the top surface of the drawing table. 
A sheet of paper may now be put in place of the horizontal 
plate of glass, and it will represent the revolved position of 
the planes precisely as they are in the drawing. • 

In the descriptions of the problems, G L refers to the ground 
line, or intersection of the planes of projection, and G' L' refers 
to the line of intersection of the vertical plane of projection 
with the plane on which the body rests. 



24: 

The plates must be lettered to sliow the general title, and 
the numbers of the problems, as shown in Plate VI. The let- 
ters used in describing the problems, however, need not be 
drawn. 

All lines that would not be seen from the position indicated 
bj' the projection in question, niQst be indicated b}' fine dots. 
The problems in the finished plates must be shaded according 
to the directions above. 

PROBLEMS IN PROJECTION. 

Problem 1. To construct the projections of a prism one and 
a quarter inches square at the base and two and a quarter inches 
in height, of whose faces one rests on a horizontal plane, and 
one is parallel to the vertical plane of projection. Draw the 
square, A B C D, equal to the top face of the prism, above Gr 
L, with C D one-quarter of an inch from G L and parallel to it. 
Draw from C and D lines perpendicular to Gr L, and prolong 
them below until the}^ intersect G' L'. Measure off the 
height of the i^rism from G' L', and draw a horizontal line for 
the top line of the elevation. The rectangle, E F G H, formed 
below the line last drawn will be the elevation of the prism, 
and the square above G L will be the plan. No dotted lines 
will appear in this problem, as all the lines on the opposite 
sides of the object will be covered b}' the full lines in front. 
Shade accordino^ to the directions above. 

Proh. 2. To revolve the prism of Problem 1 through a given 
angle about an edge through H, so that the planes parallel to 
the vertical plane shall remain so. 

Locate H on G L as far to the right in its rectangle, as H, in 
Problem 1, is from the border line. As the revolution has 
been parallel to the vertical plane, the elevation will be un- 
changed in form and dimensions, but will be inclined to G L. 
Lay ofP G H, making the given angle of revolution with G' L'. 
Complete E F G H, on G H as base. Since the body has 
revolved parallel to the vertical plane, the horizontal projec- 
tions of lines perpendicular to the vertical plane as A C and B 
D, have not changed in length, but those of lines parallel to the 
vertical plane, as A B and C D, will be shortened. 

Considering these facts, and that the two projections of a 
point are in the same line perpendicular to G L, the following 
is seen to be the construction of the plan: 



25 

Prolong A B and C D of Problem 1, indefinitely to the 
right. As E F G H represent the same points in both eleva- 
tio]is, erect perpendiculars from each of these points in Prob- 
lem 2j intersecting the indefinite lines just drawn, for the 
plans of the same points- A B C D and I J K L will be the 
bases of the prism in its revolved position. K L is to be dotted 
because not seen. Shade as directed above.* 

Proh. 3. To revolve the prism, as seen in its last position, 
through a horizontal angle, that is about a line through H, 
perpendicular to the horizontal plane. 

As the revolutionis parallel to the horizontal plane, the plan 
is changed only in position, not in form or dimensions. 

Therefore, draw the plan of Problem 2 inclined to G L at the 
angle of revolution, taking L in L D produced from Problem 2. 

Now as each point of the bodj^ revolves in a horizontal plane^ 
its vertical projection will move in a straight line parallel to G 
L. Hence we make the following construction for the eleva- 
tion: In the case of any point, as B, in the plan, draw a per- 
pendicular from this point to G L, and from F, which is the 
elevation of B in Problem 2, draw an indefinite line parallel to 
G L. The intersection of these lines gives the elevation of the 
point in its revolved position. Proceed in the same way with 
all the other points. 

It will be noticed that in the three positions of the body just 
taken the plan is drawn first in Problem 1, the elevati(»n first 
in Problem 2, and the plan first in ProbJem 3. The reasons 
for so proceeding are evident from the constructions. 

This order will hold true in all the problems following. 

Proh. 4 To construct the projections of a square prism of 
the same size as in Problem 1, whose lower base is on a horizon- 
tal plane, and whose lateral faces make angles of forty-five 
degrees with the vertical plane. 

Lay out the plan, A B C D, a square, with its sides at forty-five 
degrees to G L, and with C one-quarter of an inch above it. 
Draw perpendiculars from its vertices to G L, and prolong 
them below G L, as in Prob. 1, for the vertical edges in the 
elevation. Join the extremities of these equal vertical lines. 

*Nothmg more will be said about Bhaclius, but it is to be understood that each i)rojoc- 
tiou is to be shaded, with the light taken from over the left shoulder at an angle of fortj'- 
five degrees with the horizontal plane. The shading is a very important point in these 
problems. 



26 

Proh. 5. To revolve the prism of Prob. 4, through a given 
angle, about the lower right-hand corner, so that the vertical 
edges shall revolve in planes parallel to the vertical plane of 
projection. 

In this problem, as in Prob. 2, the revolution will be such 
that the vertical projection will be changed only in position. 
La}' off, then, E F, making with G L the given angle of revo- 
lution, and draw the elevation of Problem 4 on E F as a base. 
Draw the plan, as in Problem 2, by erecting perpendiculars 
from each point in the elevation, and by drawing indefinite 
lines from corresponding points in the plan of Problem 4. The 
intersection found in this way will be the points in the plan. 
The revolution being the same as in Problem 2, the reasons 
given there apply to this case as well. 

Prob. 6. To revolve the prism of Problem 5 through a given 
horizontal angle about E. 

Draw the plan of Problem 5, inclined to G L, at the given 
angle of revolution locating the plan of the point E, on the 
same horizontal line as it is in Problem 5. Gret each point on 
the elevation, in the same way that the points in Problem 3 
are found, by finding the intersections of perpendiculars and 
horizontals, drawn respectively from the plan, and from the 
elevation of Problem 5, remembering that the points revolve 
horizontally, and do not change vertically. 

The projections of the point A in Problems 4, 5 and 6, are 
indicated by the dotted lines on the cuts. The other points in 
Problem 6 are found in precisely the same way. 

Proh. 7. To construct the projections of a regular hexagonal 
pyramid, and the projections of a section of that pyramid 
made by a plane which is perpendicular to the vertical plane. 

The height of the jDyramid is to be the same as that of the 
prism in Problem 1, and the diameter of the circumscribing 
circle of the base is to be two inches. 

The part of the pyramid above the section is to be represented 
by dotted lines and the lower part, or frustum, in full lines. 

To find the projections of the pyramid, draw a regular hex- 
agon, A B C D E F, above Gr L, with lines joining the oppo- 
site vertices for the plan of the pyramid; draw perpendiculars 
from the vertices to G L. The intersections of these perpen- 
diculars with G L will be the elevations of the corners of the 
base. Erect a perpendicular from the center of the elevation 



27 

of the base, and on it measure off the vertical height to H, 
join H with the points on the base for the elevations of the 
edges. 

To draw the projections of the section, assume L K making 
an angle with G L, and cutting the pyramid as shown in the 
plate. Draw M N perpendicular to G L, and vertically above 
L. K L represents the intersection of the cutting plane with 
the vertical plane; and M N, its intersection, with the horizon- 
tal plane. 

These lines are called the traces of the cutting plane, and 
must be represented by broken lines like G L. The elevation 
of the section will be the part of K L included between the 
limiting edges of the pyramid, and is to be shown a full line. 

The plan of the section is found by erecting perpendiculars 
to G L from the points where K L cuts the elevations of the 
edges of the pyramid, and by finding where these perpendicu- 
lars intersect the plans of the same edges. T represents the , 
plan and elevation of one point in the section. 

Prob. 8. To revolve the frustum of the pyramid in Prob- 
lem T through a vertical angle about J. 

Draw J I equal to the same line in Problem 7, and making 
a given angle with G' L\ Construct, on J I, an elevation like 
the one in Problem 7. 

Each point in the plan may be found, as in the second posi- 
tions of the prisms, by erecting perpendiculars from the points 
on the elevation, and finding their intersections with horizon- 
tals drawn from the plans of the same points in Problem 7. 
T shows the plan and elevation of the point T in Problem 7. 

Prob. 9. To revolve the frustum of Problem 8 through a 
given horizontal angle. 

Draw the plan like that of Problem 8, but making the given 
angle of revolution with G L. 

Each point in the elevation may be found by drawing per- 
pendiculars and horizontals respectively, from points in the 
plan, and from corresponding points in the elevation of Prob- 
lem 8. The revolution of the point T is indicated by the dot- 
ted lines. 

Prob. 10. To construct the projections of a regular octa- 
gonal prism, one of whose bases is in a horizontal plane. 

Lay out a regular octagon, one inch and three-quarters 
between parallel sides, for the plan; and from its vertices draw 



28 

verticals to G L, and produce them below G L, until they 
intersect G' L" , for the vertical edges of the prism. Make the 
top line of the elevation three-eighths of an inch below G L. 

Pivb. 11. To revolve the prism of Problem 10, parallel to 
the vertical plane, through a given angle. 

Construct, as usual, the vertical projection, differing only in 
position from that in Problem 10. In the case of any point 
as T, to find its plan, erect a perpendicular to G L from the 
elevation of the point and draw a horizontal from the corres- 
ponding point in the plan of Problem 10. The bases of the 
prism in this position will be equal octagons though not reg- 
ular. 

Prob. 12. To revolve the prism of Problem 11 through a 
given horizontal angle. 

Draw the plan like that in Problem 11, making the given 
angle of revolution with G L. In the case of each point, to 
obtain the elevation^ drop a vertical from the plan of the point, 
and draw a horizontal from the corresponding point in the 
elevation of Problem 11. The bases in the elevation will be 
equal octagons, in which the parallel sides are equal. 

The point T is the same point that is marked in Problems 
10 and 11. 

Prob 13. To construct the projections of a given right cir- 
cular cylinder, whose lower base is in a horizontal plane. 

Draw a circle of the given radius, fifteen-sixteenths of an 
inch, for the plan. The elevation of a right cylinder in this 
position is evidently a rectangle. 

Drop perpendiculars from the right and left hand limits of 
the plan for the limiting elements in the elevation. Make the 
elevation of the same height, and the same distance below G 
L, as in Problem 10. 

Prob. 14. To revolve the cylinder of Problem 13 through a 
given angle parallel to the vertical plane. 

The elevation will be a rectangle inclined at the angle of 
revolution. In the same way that the regular octagon in Prob- 
lem 11 was shortened in one direction, remaining no longer 
regular, the circle in Problem 13 will be shortened horizontally 
and we shall have an ellipse. 

This ellipse is constructed b}^ points. Returning to Problem 
13, divide the semi-circumference above the horizontal diam- 
eter into any number of equal parts. It is not absolutely 



29 

essential that these parts he equal; but, for convenience, they 
are usually so taken. Drop verticals from each of these points 
of division intersecting the semi-circle below and also the bases 
of the cylinder in the elevation. Returning to Problem 14, 
space off on the two bases in the elevation divisions equal to 
those on the corresponding bases in Problem 13. 
, To get, now, the plan of either base, erect verticals from the 
points in the elevation of that base, and draw horizontals 
from the two points corresponding in Problem 13. 

The point marked T in problem 14 is the second point from 
the right in Problem 13. 

It will by observed that in this case each point in the eleva- 
tion corresponds to two points in the plan. There will evi- 
dently be the same number of lines across the plans of the bases 
in Problem 14, as there are in Problem 13, and they will be of 
the same length in both cases, but slightly nearer together in 
Problem 14 than in Problem 13. To complete the plan, after 
getting the bases, draw horizontals tangent to the two bases 
above and below. These will be the limiting elements of the 
cylinder in the plan, but will correspond to no line given in the 
elevation. 

Proh. 15. To revolve the cylinder of Problem 14 throuo^h a 
given horizontal angle. 

Draw the plan of Problem 14, making the given angle of 
revolution with Gr L. This can best be done by drawing a 
center line in Problem 14, and then drawing a corresponding 
line making the given, angle with G L. 

Erect, then, perpendiculars to this last line equal in length 
and corresponding in position to the lines crossing the bases 
of Probleml 4. The ellipses will be exactly like those in Prob- 
lem 14, For the elevation, proceed as in Problem 12, by drop- 
ping verticals from points in the plan, and finding their inter- 
sections with horizontals from corresponding points in the 
elevation of Problem 14. The construction lines for the point 
T are given in the plate. The two bases in the elevation will 
be ellipses, projections of the ellipses in the plan. Draw the 
limiting elements tangent to these ellipses at the right and 
left. 

Proh. 16. To construct the projections of the frustum of a 
right cone, whose base is in a horizontal plane. 

Draw a circle of the same radius as in the cylinder above for 



30 

the plan of the base. Drop verticals from the right and left- 
hand limits of this circle intersecting G' L', which should be the 
same distance below G L as in problem 13, for the elevation of 
the base. Drop another vertical from the center of the circle, 
and on it measure the same height from G' L' as that of the 
C3'linder above. This will give the elevation of the apex of the 
complete cone, the plan of which is the center of the circle 
above. Join this point with the two ends of the elevation of 
the base for the limiting elements in the elevation. 

The upper base of the frustum in this case, is formed by a 
a plane, cutting the cylinder, perpendicular to the vertical 
plane of projection, and making an angle with the horizontal 
plane. The cutting plane is given by its traces on the plate. 
This upper base will be an ellipse, as is every section of a cone 
made by a plane that does not cut the base of the cone. 

The elevation of this upper base will evidently be that part 
of the vertical trace of the cutting plane included between the 
limiting elements. 

To get the plan of this base^ proceed as follows: Divide the 
straight line, representing the elevation of this base, into any 
number of equal parts, and through these points of division 
draw horizontals as shown on the plate. 

The distances of these points from the axis of the cone are 
evidently equal to the lengths of the horizontals drawn through 
these points in the elevation, and limited by the axis and the 
limiting elements. Hence, to get any point, like A, in the 
plan, erect a vertical from the elevation of that point; and 
with the plan of anj point in the axis, as a center, and 
the horizontal M x^ through the point in the elevation as a 
radius, describe arcs intersecting the vertical, A and the point 
above it, are both found by using the same radius, getting the 
intersections above and below with the vertical from the point 
in the elevation. 

Proh. 17. To revolve the frustum of Problem 16 through a 
given angle parallel to the vertical plane. 

Construct, as in all the preceding similar cases, the eleva- 
tion of the preceding problem, making the given ans'le with 
GL. 

Get the plan of the lower base precisely, as in getting the 
bases of the cylinder in Problem 14. 

To get the plan of the upper base, erect verticals from points 



31 

in the elevation corresponding to those marked in Problem 16, 
and find the intersections of horizontals from the plans of the 
same points in Problem 16. 

la case the frustum were turned through a larger angle 
than that shown in the plate, limiting elements would show 
tangent to the two ellipses in the plan. 

Proh. 18. To revolve the frustum of Problem 17 through a 
given horizontal angle. 

Construct the plan of Problem 17; making the given angle 
with G L, in a manner similar to that employed in construct- 
ing the plan of the cylinder in Problem 15. 

Get the points in the two bases by dropping verticals from 
the points in the plan, and finding the intersections of hori- 
zontals from corresponding points in the elevation of Problem 
17. Draw the limiting elements tangent to the ellipses found 
in this way. 

The orthographic projections of an ellipse are always ellipses, 
the circle and the straight line being special cases of el- 
lipses. 

PRACTICAL APPIilCATIONS OF PROJECnON 

Roof Truss. — * Scale one half an inch to the foot. 

In this case, the two sides being symmetrical, an elevation of 
one half and a section through AB, will show every part, and 
are therefore chosen as the bast views for the working 
drawing. 

First, lay ofi" the dotted horizontal line B C, which is twenty - 
f^'VQ feet in length. Then at E erect the vertical D E making 
E B eight feet and E D nine feet, reducing each to the proper 
scale. 

Join C and D. These lines are the center lines of the 
principal timbers in the truss and of the iron rod D E. 

Draw next the timber of which C B is the center, fifteen 
inches deep, leaving the end near C unfinished until the rafter 
is drawn. The other two timbers of which the center lines 
have been drawn are twelve inches deep. Draw the lines paral- 
lel to the center lines. The line forming the joint at D is 

* The Scale of a drawing ie the ratio that the lines on the drawing bear to the actual 
lengths of the lines on the object. The Scale should always oe stated on a drawing; and 
may be given as a fraction like 14, M etc., or it may be stated as a certain number of 
inches to the foot. 



32 

found by joining the intersections of tlie lines of the timbers. 
The joint above C is formed b}' cutting in three inches in a 
direction perpendicular to the upper edge of C D and joining 
the end of this perpendicular to the intersection of the lower 
line of C D with the upper line of C B. 

The hatched pieces near C, H and D, are sections of the 
purlins, long pieces resting on the truss and supporting the 
rafters. These purlins ai e cut into the rafters and into the 
truss one inch, with the exception that the one near C is not 
cut into the truss. Those should next be drawn, rectangles 
ten inches hy six inches. The center line of the one near C 
is a prolongation of the short line of the joint at C, the one 
near D is three inches below the joint at the top, and the 
third is halfway between the other two. 

The center line of H E should start from the lower part of 
the center line of the middle purlin, and the top edge should 
meet the top edge of C B in D E. With one half the dei^th of 
H E, four inches, as a radius describes an arc with the point 
where D E meets the upper line of C B as a center. Draw the 
center line through the point indicated above, and tangent to 
this are. Cut in at E one third the depth of H E, or two and 
two-thirds inches and, at H, one inch, in the way shown 
in the plate. 

Draw next the rafter, F A. twelve inches deep and eight 
inches from the top of C D. Cut into the rafter a horizontal 
distance of six inches for the end of the beam C B, and make 
the end of the rafter in line with the bottom of C B. 

Make a short end of the rafter on the other side as shown at 
A, and show the horizontal pieces broken off as shown in the 
plate. Make the rod, D E, an inch and a half in diameter with 
washer and nut at the lower end, and with a head and an angle 
plate running to the purlin at the upper. Make the washer 
and angle plate one inch thick, the washer four inches in di- 
ameter and the nut and head according to the standard. The 
short bolt near E is one inch in diameter and the head, nut 
and washer of two-thirds the size those on in D E. The 
short bolt near C is precisely like the one just described, make 
the angular washers at the bottom of the s'dine diameter and 
at right angles to the bolts as in the other washers. 

The sectional view at the right is formed by projecting lines 



f 



33 

across from the elevation just drawn and measuring oiF the 
proper widths. The timbers of the truss proper are all twelve 
inches wide, the rafter is three inches wide and the purlins 
are broken off so as to show about two feet in length of each. 

Great care must be taken in inking not to cross lines, those 
nearest the observer in any view should be full lines, and those 
hidden by them should be broken off or dotted. 

Hatching, as it is called, is a method of representing a surface 
cut, as in a section, and it is done by drawing fine parallel lines 
at an angle of forty-five degrees with the vertical and very 
near together. 

In case two different pieces joining are cut, the lines should 
be at right angles to distinguish the two sections. There are 
no such cases in this example however. It is very important 
that a hatched surface shall look even, and this can only be 
effected by making all lines of the same width, and the same 
distance apart. 

The title, which is given in italics at the beginning of this 
description, may be placed within the border as indicated by 
the dotted lines. 

The student will be allowed to choose any raechanical letters 
for the title, but the heights must be three-eighths of an inch in 
the words Roof Truss, which must be in capitals, and the letters 
in the words indicating the scale are to be one-half as high. 
The scale should be put on thus : Scale J'' = 1', one dash indi- 
cating feet, and two dashes inches. 

Stub End of a Connecting Rod. 

The two projections chosen to represent this object are a 
front elevation, and a longitudinal section through A B. In 
this case these projections show the different parts much more 
clearly than they could be shown in plan and elevation. 

The section at the right of the front elevation shows what 
would be seen from that position were the part on the right of 
A B removed. The set screw that holds the key is shown as 
though not cut through A B. This example illustrates the 
necessity of hatching to distinguish the cut portions from those 
beyond. It also shows the proper method of representing 
the different pieces, shown in section, by lines running in dif- 
ferent directions on adjacent pieces. 

The dimensions to be used on the full sized drawing are 



34 

marked in inches on the cut. The arrow heads on either side 
of the dimension marked represent the limits of the dimension. 
It will be noticed that some of the dimensions at the top are 
diameters, whilst others are radii. 

'• Divide the space within the border on a half-sheet of Imper- 
ial by two vertical lines making three equal spaces. Use these 
lines as center lines of the two projections. Draw the front 
elevation first. Commence by assuming on the line A B, the 
center of the inner circle, near the top, and describe the two 
concentric circles, about this center with the given radii- The 
common center of these circles should be taken far enough 
below the border at tho top to leave about the same amount of 
margin above and below when the elevation is completed. 
Take a point on A B, five-sixteenths of an inch above the first, 
and about it as a center, describe the semicircle of the upper 
limit of the brass bushing, using a radius of one inch and a 
half. From this same center describe the dotted semicircle 
with the radius indicated. One-half au inch above the first 
point on A B, take a third point as the center of the semicircle 
of the top of the ptrap. 

Notice, in the cut, what lines are tangent to the semicircles 
just drawn, and proceed with the drawing, making each 
dimension as indicated. No dimensions are marked on the 
irregular curve near the bottom, which shows the intersection 
of the flat face of the rectangular end of the rod with the cyl- 
indrical part below; but such an unimportant line does not 
require to be so carefully drawn to dimensions as the others, 
and it is sufficient to draw it as nearly as possible like that in 
the cut, taking great care to have the two sides exactly 
alike. The horizontal lines in the section may all be projected 
from corresponding points in the front elevation. All the hor- 
izontal distances are indicated on the section, but the vertical 
distances, being the same as in the front elevation, are pur- 
posely omitted. The most difficult part of the work in this 
drawing is to make the hatching even. Use a sharp pen and 
make all the lines of the same width and the same distance 
apart. 

Projections of Screws, 

The thread of a screw may be considered to be generated by 
a section moving uniformly around a cylinder, and at the same 



35 

time uniformly in a direction parallel to the axis of thecylin- 
der. Plate X shows the true projections of a Y-threaded screw 
at the left, and of a square threaded screw at the right. 

V- Threaded Screw. — Commence by describing a semicircle 
with a radius of one inch and a half, as shown in the outer dot- 
ted circle in the plan. This will be the half plan of the outer 
part of the thread. Drop verticals from the outer limits of the 
semi-circle for the limiting lines of the V threads in the eleva- 
tion. The projections of the head and an outline of a section 
of the nut should next be drawn. The standard dimensions of 
heads and nuts are expressed by the following formulae, in 
which d is the oufcside diameter of the screw, h the thickness of 
the head or nut and D the distance between the parallel sides 
of the head or nut : D = li d + i", h = D -- 2. 

Construct the projections of the heads and nuts according 
to this standard, and show the hexagonal head finished as in 
the plate. The short arcs that cut off the corners are described 
with the middle of the lower line of the nut as a center, and 
the longer arcs bounding the top faces of the head are described 
with the middle point af the lower line of each face as a cen- 
ter. The top of the head is a circle, as shown in the plan. 

A section of a standard V-thread is an equilateral triangle 
all the angles of which are sixty degrees, hence the outlines of 
the sides of the elevation may be drawn by means of the thirty 
degree triangle used on the T-square. 

Before drawing these triangles, however, the pitch must be 
determined. The pitch of a screw is the distance from any 
point on a thread to another point on the same thread 
on a line parallel to the axis. The pitch is usually expressed 
by stating the number of threads to the inch. This screw 
has two threads to the inch, therefore the pitch is one-half 
an inch. This represents the advance in the direction of the 
axis during one revolution. 

Lay off, then, on the'limiting line at the left, distances of 
one-half an inch, commencing at the bottom of the head. 

Through these points draw lines as indicated above, making 
a series of triangles. The inner intersections of these lines will 
be in a vertical line, which, projected up, gives the radius of 
the inner dotted semicircle in the plan. This semicircle is a 
half plan of the bases of the threads. As the thread advances 



36 

a distance equal to the pitch in a whole revolution, it is evident 
that in a half revolution the advance will be equal to half the 
pitch; therefore commence on the right hand limiting line with 
the first space a quarter of an inch, and from this point on, 
make the spaces equal to the pitch. Describe a series of 
triangles on this side in the same way as before. 

Every point in the generating triangle describes a helix as 
it revolves about and at the same time moves in the direction 
of the axis of the screw. It is evident that the helices described 
by the vertices of the triangle will be the edges and intersec- 
tions of the threads. The manner of getting the projections 
of these lines will now be described. The plans of these hel- 
ices will be the circles which have just been obtained and 
which are shown in the plate in dotted lines. Draw from the 
outer vertex of one of the triangles representing the edges of 
the threads, an indefinite line toward the left as shown in the 
plate. Divide the semicircle above into a number of equal spaces, 
eight at least, and draw radii to these points of division. Lay 
oif the same number of equal divisions on the indefinite line, 
and at the last point erect a perpendicular equal in length to 
one-half the pitch. Join the end of this line with the right hand 
end of the horizontal line, forming a triangle. Erect verticals 
from each point of the division of the horizontal line. To 
find any point, like A, in the helix forming the edge of the 
threads, drop a vertical from one of the divisions of the semi- 
circle, and find where it intersects a horizontal drawn from a 
corresponding point on the diagonal line of the triangle at the 
left, counting the same number of spaces from the right on the 
diagonal line as the point taken on the semicircle is from the 
left. As many points may be found this way as there are on 
the semicircle. Join these points by using the irregular curve. 

The points in the helices at the bases of the threads may be 
found in the same way as shown by the dotted lines, the equal 
divisions of the semicircle in this case being where the radii of 
the center semicircle cut this one. The reason for this construc- 
tion will be plain on considering that the equal spaces on the 
arc, represent equal angles of revolution of the generating 
triangle; and the distances between the horizontals drawn 
from the points of division of the diagonal line, represent the 
equal rates of advance in the direction of the axis. 



37 

As the curves at the edges of the dijfferent threads are all 
alike, a pattern should be made, from thin wood, of the one 
constructed, and this should be used to mark all the long 
curves of the screw and nut. Another curve should be made 
for the inner helices. 

The helices will evidently be continuous from one end of the 
screw to the other, but the dotted lines which would show the 
parts on the back side are left out in order that the drawing 
may not be confused bj^ too many lines. 

In the plate, the screw is shown as entering only a short dis- 
tance into the nut which is shown in section below. 

The threads of the nut are the exact counter parts of the 
threads of the screw; but as the threads on the back side of 
the nut, only are shown in the section, the curves run in the 
opposite direction. A small cylindrical end is shown on the 
bottom of the screw. This represents the end of the cylinder 
on which the thread is wound. ' 

Square Threaded Screws. The square threaded screw is 
generated by a square revolving about the cylinder and at the 
same time moving in a direction parallel to the axis. In the 
square single threaded screw the pitch is equal to the width of a 
space and the thickness of a thread, measured in a direction 
parallel to the axis. 

Draw the projections of the head and nut of the same dimen- 
sions as in the Y-threaded screw. Laj^ off a series of squares, 
the sides of which are equal to one-half the pitch, on the two 
edges of the screw, and find the points in the helices as in the 
example preceding. It should be observed that the long curves 
show in their full lengths, and the short ones only show to the 
center in the screws, whilst in the nut the opposite is true. 

The smaller screws near the center of the plate, show how V 
and square threaded screws are after represented when so 
small that the construction of the helices is impracticable. 
The construction only varies from the larger ones inasmuch 
as the curves are replaced by straight lines. 

Below there is shown still another method of representing 
very small screws, either V or square threaded, and the projec- 
tion of a hexagonal head with face parallel to the plane of 
projection. 



38 

Below is given a table of the Franklin Institute, or United 
States standard proportions for screw threads. This table is 
given here that it may be conveniently referred to whenever 
screws and nuts are to be drawn. A real Y thread is often 
used, but a thread very similar, having a small flat part, in 
section, at the outside of each thread and a similar flat part 
between the threads, is becoming more common. The dimen- 
sions of such a thread are given in the following table, where 
diameter of screw means the outer diameter, diameter of core, 
the diameter of the cylinder on which the thread is wound, and 
width of flat, the width of the flat part just described. The 
four columns at the right relate to the nuts and bolt heads. 



39 



PROPORTION OF SCREW THREADS, 
NUTS li BOLT HEADS. 





SCZEeETTT^S. 




liTTTTS ^^niTTD lEIE^i^lDS. 










HEXAGONAL. 


SQUARE. 


1 


DIAMETER 


THREADS 


DIAMETER 


WIDTH OF 


OUTSIDE DI- 


INSIDE DI- 




HEIGHT OF 


OF SCREW. 


PER INCH 


OF CORE. 


FLAT. 


AMETER. 


AMETER. 


DIAGONAL. 


HEAD. 


1 


20 


.185 


.0062 


9 
16 


1 
2 


1 1 
16 


1 

T 


5 

T6 


18 


.210 


.0070 


1 1 
16 


1 9 
33 


1 3 

T6 


1 9 
6T 


1- 


16 


.291 


.0078 


25 
3T 


11 
16 


u 


ii 


7 
T6 


U 


.311 


.0089 


43 

T8 


25 

33 


IrV 


25 

6¥ 


1 


13 


.400 


.0096 


1 


¥ 


li 


7 
16 


9 
16 


12 


.451 


.0104 


leT 


31 
3^ 


m 


fi 


5 

"8 


11 


.507 


.0113 


laV 


ItV 


li 


1 7 
33 


3 


10 


.620 


.0125 


ItV 


li 


11 


5 

8 


7 
8 


9 


.731 


.0140 


111 

-L32 


ItV 


23V 


23 
32 


1 


8 


.837 


.0156 


J-8 


If 


2A 


13 

T6 


li 


r 


.940 


.0180 


2A 


lit 


2i 


29 
6¥ 


li 


7 


1.065 


.0180 


2-1% 


2 


211 


1 


If 


6 


1.160 


.0210 


21 


-^16 


3tV 


IsV 


li 


6 


1.284 


.0210 


21 


93 


31 


lA 


11 


. 5i 


1.389 


.0227 


Ql 5 

^16 


9_9_ 
-i/16 


31 


lA 


If 


5 


1.490 


.0250 


3 re 


21 




If 


11 


5 


1.615 


.0250 


3if 


2il 


4t\ 


lit 


2 


tti 


1.712 


.0280 


31 


3i 


4T^,r 


1/6 


21 


-li 


1.962 


.0280 


4tV 


3i 


J^3 1 


If 


2i 


1 


2.175 


.0310 


4i 


Si 


5i 


HI 


21 


1: 


2.425 


.0310 


411 


41 


6 


2i 


3 


3i 


2.628 


.0357 


51 


41 


6A 


2A 



40 

DRAWING FROM ROUGH SKETCHES. 

Plate XI is given to illustrate the method of making rough 
sketches of an object from which a finished drawing is to be 
made. The rough sketches here shown are of a large valve 
such as is used on large water pipes.* This example has been 
chosen because it is symmetrical with reference to the center 
line. In such a case as this, it is obviously unnecessary to 
make complete sketches of the whole object. Enough of the 
plan of the object is given above to make a complete plan from, 
in the drawing. The sketch of the elevation below shows all 
that is necessary for that. 

In making a rough sketch decide what projections will best 
represent the object, and get in such a position as to see the 
object as nearly as possible as it will apj^ear in the projection, 
changing the position of observation for the sketches of the 
different projections. It must be borne in mind that the view a 
person has of an object while sketching is a perspective view 
and allowance must be made for the way it will appear in 
projection.. Sketches similar to the projections of the objects 
are better than perspective sketches to work from. The 
sketches should be made in the same relative position that they 
will appear in the projection drawing. Be sure to represent 
every line of the object in the sketches, excei^ting the cases 
where symmetrical iDarts may be drawn from a sketch of one 
part, and indicate all the dimensions by plain figures and arrow 
heads, ff/A-//^g all. tlie dimensions possible from some icell defined 
line like a center line or a. bottom line. 

If any of the details on the principal sketch are too small to 
contain the figures of the dimensions make enlarged sketches 
aside from the other as indicated in the plate. All that is 
necessary to be known about a nut is the diameter of the 
bolt; the nut may then be constructed according to the 
standard. 

Often a few words of description written on the sketch as, in 
the case of a bolt, four threads to the inch, will describe a part 
sufficiently to one acquainted with the standard proportions of 
such common pieces as screw bolts, etc. 

*A complete drawing of such an object should show the internal parts, but as the object 
of this plate is simply to illustrate the method of sketching, the internal arrangement is 
not ?hown. * 



41 

One unaccustomed to making sketches, is apt to omit some 
dimensions, and too great care cannot be taken to have 
every part of the object clearly indicated in some way on the 
sketch. 



TINTING AND SHADING. 



At this point of the work the following additional materials 
will be needed: 

A set of -water colors, a nest of cabinet saucers, a camels hair brusli, 
a bottle of mucilage and brush, and a small glass for -water. 

Watek Coloes. — Winsor & Newton's water colors in "half- 
pans" are recommended . The set should contain the following 
colors: — Burnt Sienna, Raw Sienna, Crimson Lake, Gamboge, 
Burnt Umber, Indian Red and Prussian Blue. If the bottled 
ink has been used for the previous work, a stick of India ink 
will also need to be purchased. All the conventional colors 
used to represent the different materials may be mixed from the 
simple ones given in this list. 

Saucees. — The "nest" should contain six medium sized 
saucers. 

Beush. — The camel hair brush should have two points^ and 
should be of medium size. 

Mucilage. — The mucilage needs to be very thick, as it is 
used in shrinking down the heavy drawing paper- The ordin- 
ary mucilage in bottles is not fit for this use, and it is recom- 
mended that each person buy the Grum Arabic, and dissolve it 
in a bottle of water, using it as thick as it will run. 

Watee Glass. — This glass is for holding clean water with 
which the colors are mixed. Any small vessel will answer this 
purpose, but a small sized tumbler is the most convenient. 

DIRECTIONS FOR SHRINKING DOW^N PAPER. 

Whenever a drawing is to be tinted it must be shrunk down 
in order that it may not wrinkle after tinting. To shrink 
down the paper proceed as follows: Lay the edge of the T 
square parallel to an edge of the paper, and about five-eighths 
of an inch from it; and turn up the paper at right angles, mak- 
ing a sharp edge where the paper is bent up by pressing it 



42 

hard against the edge of the T square with the thumh nail or 
a knife blade. Turn up all the edges in this way so that the 
paper will resemble a shallow paper box. The corners need 
not be cut, but must be doubled over so that all the edges 
of the paper will stand nearly perpendicular. After this is 
done the paper should be turned over so as to rest on the up- 
turned edges, and dampened very slightly with a sponge on 
the back. Every part of the paper must be dampened except 
the upturned edges which must be kept dry in order that the 
mucilage may stick. No water should be left standing on the 
sheet when it is turned over. 

The paper should next be turned over and placed so that two 
edges at right angles may correspond when turned down to 
two edges of the drawing board. The other side should then 
be thoroughly wet. The mucilage should next be applied to 
the dry edges as rapidly as possible. The two edges that cor- 
respond to the edges of the drawing board should first be turned 
down, great care being taken to leave no wrinkles in these 
edges nor in the corner between them. The other edges should 
then be turned down, the same care being taken to leave no 
wrinkles either in the edges or corners. The edges must be 
kept straight, and, if there are no wrinkles left in the edges, 
the paper will come down smooth when dry, no matter how 
much wrinkled while wet. The natural shrinkage of the paper 
is sufficient without stretching. The edges should be pressed 
down smooth with the back of a knife or the thumb nail, and 
the paper should be allowed to dry slowly. 

Considerable practice may be necessary before the paper can 
be shrunk down successfully, but if the directions above are 
followed closely there need be no difficulty. The paper must 
be dampened evenly, and the mucilage must be put on evenly 
and abundantly. Great care must be taken not to drop any 
mucilage on the middle of the drawing board, and not to get 
any beyond the dry edge of the paper. Otherwise the paper 
may be stuck down so as to make trouble in cutting the plate 
oflp when finished. 



T HE PLATES IN TINTING AND SHADING 

Plates A and B, on the walls of the Mechanical Drawing 
Room at the University, contain the most common forms that 



43 

are brought out by shading in ordinar}^ mechanical drawings 
and the most common conventional colors used in working 
drawings.* 

Plate A, which is shaded altogether with India ink is to be 
done first. 

Shrink down a half sheet of Imperial paper, and mark it 
inside of the edges so that it may be, when cut off, twenty by 
thirteen and a half inches. Lay out a border one inch in- 
side of the lines just drawn and draw the outlines of the figures 
with a very sharp pen, using the best of ink, and making the 
lines as fine as possible. The figures must be drawn of the 
same size, and arranged in the same way as in the wall plates. 
The border and the letters, which are are to correspond with 
those in the wall plates, should not be drawn until the plate is 
shaded. The dimensions of the figures need not be put on to 
the finished plates at all. 

After inking in the figures, the plate should be washed over 
with clean water to take out any surplus ink and to leave the 
paper in better condition for the water shades. The paper 
should be sopped very lightly with a sponge and a large quan- 
tityof water should be used. After washing the paper allow 
it to dry slowly. If the paper is dried in the sun it will get so 
warm that the shades will dry too rapidly. When the paper 
is down smooth and dry, it should be placed on the drawing 
table slightly inclined in one direction in order that the ink or 
water color may always flow in one direction. 

Take a saucer half full of clean water and by rubbing the 
wet brush on the end of a stick of ink mix enough India ink 
to make a shade no darker than that in (c), on wall plate. A 
small piece of paper should be kept to try the shades on before 
putting them on the plate. Mix the ink thoroughly with one 
end of the brush before applying to the paper. One end of the 
brush should always be used in the ink or tint while the other 
end is kept clean for blending. 

*These colored plates could not conveniently be placed in this pamphlet, and in cases 
where access cannot be had to the wall plates here mentioned, it is recommended that 
the instructor make similar ones for the use of the stiidents. Plate A contains five rect- 
angles in the upper row the first three of which are to be plain shades, and the other two 
are to be blended. The lower row of figures in this plate contains plans and elevations 
of the following figures in the order named, a prism, pyramid, cylinder, cone and sphere. 

Plate B contains, in the upper row, circular figures tinted to represent the conventions 
for cast-iron, wrought-iron, steel and brass ; and in the lower row four square figures with 
the conventional colors for copper, brick, stone and wood. 



44 

With considerable ink in the brush but not nearly all it will 
hold, commence at the top line of (a.), and follow it carefully 
with the first stroke. Before the ink dries at the top, laj^ on 
the ink below by moving the brush back and forth, using 
enough ink in the brush so that it will flow gradually, with 
the help of the brush, toward the bottom. The lines must be 
followed carefully at first, and the brush should not be used 
twice over the same place. In following a line with the brush 
get in such a position that the forearm will be perpendicular to 
the direction of the line^ Do not paint the shades on but 
allow them to flow quite freely after the brush. In shading 
or tinting there is great danger of making clouded places and 
*'water lines" unless the greatest of care is taken in using the 
brush. If the brush is used over a shade that is partly dry it 
will make it clouded. And if the edge of the shade is allowed 
to dry before finishing, a ^'water line ' is produced where the 
new shade is joined to the old. 

In finishing up a figure the ink should be taken up with the 
brush so that it will not spread beyond the lines. The sun 
shonld never be allowed to shine on the paper, as it will dry it 
too fast. A damp day is better for tinting or shading than a 
dry one for the reason that the drying is then very slow. The 
shades of (<^), (h) and (c) are all plain. Commence on (a), and 
while it is drying put a coat on {h). To determine when a 
shade is dry look atit very obliquely, and if it does not glisten 
it is ready for another coat. Put four coats on (a), two on {h) 
and one on (c). 

Blending. — A varying shade, such as is noticed in viewing 
a cylindrical object, may be obtained by blending with India 
ink. This operation of blending is employed in bringing out 
the forms of objects, as seen in the lower figures of plate A. 

The figures {d) ^nd (e) are for practice in blending before 
applying to the solid objects below. Begin {d) by laying on 
a fiat shade about an eighth of an inch wide, using but little 
ink; and when nearly dry take the other end of the brush, 
slightly moistened in clean water, and run it along the 
lower edge of the shade blending downward. When this is 
entirely dry lay on another plain shade a little wider than the 
first, and blend it downward in the same way. Use but little 
water and lay on the shade in strips, alwaj^s commencing at the 
the top line. When finished the lower part will have had but 



45 

one coat whilst the upper part will have had several. Blend 
(c) in the same way as (d), but use narrower strips of tint in 
order to make more contrast between the top and bottom. 

Shading Solids. — When a solid object is placed in a strong 
light coming principally from one direction, a strong contrast 
will be noticed between the shades of the different portions, and 
these shades serve to reveal the shape of the object much more 
clearly than when it is placed in diffused light only. For this 
reason, as well as from the fact that the laws of the shades of 
an object in light from one direction are very simple, the shades 
in a drawing are usually made to correspond to those of a body 
where the light comes from a single window. In all cases, 
however, it is assumed that there is a certain amount of dif- 
fused light, such as is always present in a room lighted by a 
single window, aside from the strong beam of light that comes 
directly through the window. 

1. The shades of cm object are always in greater contrast 
when the object is near the eye than ivhenfar away. 

2, The lightest portion of a cylinder, cone or sphere is where 
the direct light strikes the object p)erpendicularly, and the darkest 
portion of the same is where the light strikes tangent to the object, 
the shade varying gradually between these parts. 

The facts just given may easily be proved by holding a body 
in the light and noticing the shades. 

These facts we will assume as the principles that govern the 
shading of the following objects. 

In view of the above principles the first thing to be deter- 
mined, after assuming the direction of the light, is where the 
lightest and darkest parts will be, and what parts are near to 
the observer and what parts are farthest away. In all the fol- 
lowing cases we shall assume the light to come from over the left 
shoulder, making the angle of forty-five degrees with both the 
vertical and horizontal planes of projection. 

The Prism. — By the use of the forty-five degree triangle 
on the T square, draw the arrows as shown on the plan. The 
points where these touch the plan show where the direct light 
will strike by the prism. By dropping verticals from these 
points we see that one will fall behind the elevation and one 
in front, showing that the vertical edge near the right separates 
the light from the dark portions of the prism The light will 
nowhere strike the prism perpendicularly, but it will strike 



46 

that face nearest the left the most directly of any, and it will, 
of course, be the lightest face of the prism. The front face will 
be a little darker and the right-hand face, being lighted only 
by diffused light, will be much darker than either of the other 
two. 

The plan showing only the upper base, receives light at the 
same angle as the front face, and will have the same shade, 
which should be about the same as on the plate, and not blended. 
Considering the principle that the contrast is less between 
light and shade at a distance, we know that the outside parts 
of the faces on the right and left will tend to assume nearly 
the same shade as they recede from the observer, consequently 
the light face should be blended slightly toward the right, and 
the dark face on the right should also be blended toward the 
right, making the former darker toward the outside and the lat- 
ter lighter toward the outside. 

The Pyra:mid. — The light and dark portions are found in 
the same way, and the elevation is shaded almost precisely like 
the prism. The top recedes slightl}- and the contrast there 
should be slightly less than at the bottom where it is nearer 
the eye. The faces in the plan recede very rapidly, and the 
greatest contrast must be at the top. The upper right hand 
part receives only diffused light. 

The Cyli]!n"der. — The dotted lines show on the plate the 
method of finding the lightest and darkest portions. Use the 
forty-five degree triangle on the T-square so as to draw the 
two diagonal radii as shown Where the one on the right 
cuts the lower semi-circumferencs is the darkest point and 
where the other cuts the same on the left is the lightest point. 
These points projected down will give the lightest and darkest 
lines on the elevation. . Blend quite rapidly both ways from 
the dark line and toward the right from the left hand limiting 
element. The shades near the limiting elements should be 
about alike on the two sides. 

The shade of the plan should be the same as that of the 
plan of the prism. 

The Coxe — The instructions given for shading the cylin- 
der with those given for shading the pyramid apply to this fig- 
ure. Great care must be taken to bring out the vertex in the 
plan. 

The Sphere. — It will be evident on consideration that the 



47 

darkest portion of the sphere is a great circle, the plane of 
which is perpendicular to the direction of the light; but, as this 
great circle is not parallel to the planes of projection, its pro- 
jections are both ellipses. There can evidently be but one 
point where the light can strike the sphere perpendicularly, 
and that is where the radius parallel to the direction of the light 
meets the surface. 

To find the lightest points in plan and elevation, join the cen- 
ters of the plan and elevation by a vertical, draw a diameter of 
the plan upward and to the right at an angle of forty-five 
degrees and of the elevation downward and to the right. Di aw 
a line using the forty -five degree triangle from the point where 
the line joining the centers cuts the circumference of the plan 
perpendicular to the diameter drawn in the same. Where it 
intersects the same will be the lightest point in the plan; aline 
similarly drawn gives the lightest point in the elevation as 
shown in the plate. The ellipse, which is the dark line of the 
object, crosses the two diameters drawn on the plan and ele- 
vation, just as far from the centers of each as the light points are 
from the same. These points may be laid off from the centers by 
means of the dividers. The shades of the plan and elevation 
of the sphere are exactly alike, but the position of the light 
and dark portions are diflPerent, as seen in the plate. 

Commence by laying on a narrow strip of shade about the 
form of a crescent over the dark point on the diameter and 
conforming as closely as may be to an arc of the ellipse. 
Add other and wider strips of the same general form, and blend 
each toward the light point and toward the outside. Great 
pains must be taken with this to get the correct shades and the 
two exactly alike. 

Put on all the lines and letters shown on the plate, making 
the dotted lines and arrows very fine. 

TiNTiKG. — Plate B contains the conventional tints for the 
following materials: Cast Iron, Wrought Iron, Steel, Brass, 
Copper, Brick, Stone and Wood. The square figures are three 
inches on a side, and the circular figures have diameters of the 
same length. These colors are more difiicult to lay on evenly 
than the India ink shades, but what has been said about the 
application of ink shades applies to them. Great pains must 
be taken to have the paper in good condition, and to keep the 
colors well mixed. Enough color should be mixed to finish 



48 

the figure as it is impossible to match the colors exactly. 
Wash the brush thoroughly before commencing a new figure. 
The plates are to be lettered like the wall plate, the initials at 
the bottom standing for the colors used. Below are given the 
materials to be used in each convention. The exact propor- 
tions of. these can best be found by experiment, comparing 
the colors with those on the wall plate. A number of thin coats, 
well laid on, generally look more even than when the tints are 
laid on in single coats. 

For Cast Iron, use India ink, Prussian Blue and Crimson 
Lake; for Wrought Iron, Prussian Blue and India ink; for 
Steel, Prussian Blue; for Brass, Gamboge, Burnt Umber and 
Crimson Lake; for Copper, Crimson Lake and Burnt Umber; 
for Brick, Indian Red; for Stone, India ink and Prussian Blue; 
for Wood, Raw and Burnt Sienna. The convention for the 
body of the wood is made by laying on a light coat of Raw 
Sienna, and the^ Grain is made by applying the Burnt Sienna, 
after the first is dry, with the point of the brush, blending 
slightly in one direction. 



SHADE LINING. 



Shade lining is a method of representing the shades of an 
object by a series of lines drawn on the projections so as to 
produce the same general efPect as when blended with 
India ink. This effect is produced by making the lines 
very fine and at a considerable distance apart on the light 
portions, and quite heavy and near together on the dark 
portions. This method of shading is often employed in uncol- 
ored drawings to bring out the forms of parts that might not 
otherwise be clearly understood, and often to give a drawing a 
fine finished appearance. 

^Plates C and D on the walls of the Drawing Room contain the 
figures that generally require to be shaded in ordinary drawings. 
Each of these plates is made on a quarter sheet of Imperial 
paper, and of the size shown. 

*Plate C contains the following figures in plan and elevation, three cylinders, ranging 
from one fourth of an inch to an inch in diameter; a hexagonal prism, and a concave cylin- 
drical surface. Plate D contains, in plan and elevation, a frustum of a pyramid, a frustum 
of a cone and a sphere. Each of these plates is made on a quarter sheet of Imperial, with 
considerable margin outside of the figures. 



49 

These plates are made smaller than the others for the reason 
that each figure contains a great many lines. 

It is recommended that a half sheet of Imperial paper be 
shrunk down, and that this be divided when the plates are 
finished. 

The dimensions of each figure are marked on the wall plates 
but need not be marked on the finished plates. 

The figures should all be penciled in of the dimensions indi- 
cated, and in the positions shown in the plates. The limiting 
lines of the figures to be shade lined should be drawn as fine as 
possible, and, on these practice figures, had better not be inked 
until the shade lines are drawn. 

The light and dark portions are found as in Plate A, and as 
the shades to be represented are the same as in that plate, 
reference is made to the remarks on shades in the description 
of the same. 

No description can be of so much value as a thorough study 
of the wall plates. Notice carefullj^ the gradation of the shade 
lines on each projection. In Plate C, the shade lines are all 
parallel; but in Plate D, they are neither parallel nor of the 
same width throughout. 

In Figure 8 the light lines are made full circles, with the 
lightest point in each projection as a center. The middle 
portions only of dark lines are made with the compasses, the 
ends being finished free hand. In shade lining the pen must 
be kept very sharp and the ink must run well. Put on all the 
arrows and dotted lines shown on the wall plates. 




■V' ITi 



LIBRARY OF CONGRESS 







019 945 577 1 m 



